Differentiate g(x) = cos5
(4x).
Using the chain rule, we have:
g'(x) = -sin(5(4x)) * 20 = -20sin(20x)
To differentiate the function g(x) = cos^5(4x), we will use the chain rule.
The chain rule states that if we have a function h(u), and u = f(x), then the derivative of h with respect to x is given by:
dh/dx = (dh/du) * (du/dx)
In this case, h(u) = u^5 and u = cos(4x).
Step 1: Find the derivative of h(u):
dh/du = 5u^4
Step 2: Find the derivative of u with respect to x:
du/dx = -4sin(4x)
Step 3: Apply the chain rule:
dg/dx = (dh/du) * (du/dx)
= (5u^4) * (-4sin(4x))
Step 4: Substitute u = cos(4x) back into the equation:
dg/dx = (5cos^4(4x)) * (-4sin(4x))
Therefore, the derivative of g(x) = cos^5(4x) with respect to x is given by dg/dx = (5cos^4(4x)) * (-4sin(4x)).